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G = C2×C41D4order 64 = 26

Direct product of C2 and C41D4

direct product, p-group, metabelian, nilpotent (class 2), monomial, rational

Aliases: C2×C41D4, C4218C22, C23.8C23, C22.24C24, C24.15C22, C41(C2×D4), (C2×C4)⋊7D4, (C2×C42)⋊11C2, (C22×D4)⋊5C2, (C2×D4)⋊12C22, C22.62(C2×D4), C2.10(C22×D4), (C2×C4).130C23, (C22×C4).123C22, SmallGroup(64,211)

Series: Derived Chief Lower central Upper central Jennings

C1C22 — C2×C41D4
C1C2C22C23C22×C4C2×C42 — C2×C41D4
C1C22 — C2×C41D4
C1C23 — C2×C41D4
C1C22 — C2×C41D4

Generators and relations for C2×C41D4
 G = < a,b,c,d | a2=b4=c4=d2=1, ab=ba, ac=ca, ad=da, bc=cb, dbd=b-1, dcd=c-1 >

Subgroups: 441 in 249 conjugacy classes, 105 normal (5 characteristic)
C1, C2 [×7], C2 [×8], C4 [×12], C22, C22 [×6], C22 [×40], C2×C4 [×18], D4 [×48], C23, C23 [×8], C23 [×24], C42 [×4], C22×C4 [×3], C2×D4 [×24], C2×D4 [×24], C24 [×4], C2×C42, C41D4 [×8], C22×D4 [×6], C2×C41D4
Quotients: C1, C2 [×15], C22 [×35], D4 [×12], C23 [×15], C2×D4 [×18], C24, C41D4 [×4], C22×D4 [×3], C2×C41D4

Character table of C2×C41D4

 class 12A2B2C2D2E2F2G2H2I2J2K2L2M2N2O4A4B4C4D4E4F4G4H4I4J4K4L
 size 1111111144444444222222222222
ρ11111111111111111111111111111    trivial
ρ21-1-1111-1-1-11-1-111-111-1-11-11-111-1-11    linear of order 2
ρ31-1-1111-1-11-11-111-1-1-11-111-11-1-11-11    linear of order 2
ρ41-1-1111-1-11-1-11-11-111-11-11-11-11-11-1    linear of order 2
ρ51-1-1111-1-1-1111-11-1-1-111-1-11-11-111-1    linear of order 2
ρ611111111-1-1-11111-1-1-111-1-1-1-1-1-111    linear of order 2
ρ711111111-1-11-1-111111-1-1-1-1-1-111-1-1    linear of order 2
ρ81111111111-1-1-111-1-1-1-1-11111-1-1-1-1    linear of order 2
ρ911111111-1-1111-1-11-1-1-1-11111-1-1-1-1    linear of order 2
ρ101111111111-111-1-1-111-1-1-1-1-1-111-1-1    linear of order 2
ρ1111111111111-1-1-1-11-1-111-1-1-1-1-1-111    linear of order 2
ρ1211111111-1-1-1-1-1-1-1-1111111111111    linear of order 2
ρ131-1-1111-1-11-1-1-11-111-111-1-11-11-111-1    linear of order 2
ρ141-1-1111-1-1-111-11-11-11-11-11-11-11-11-1    linear of order 2
ρ151-1-1111-1-1-11-11-1-111-11-111-11-1-11-11    linear of order 2
ρ161-1-1111-1-11-111-1-11-11-1-11-11-111-1-11    linear of order 2
ρ1722-22-2-2-22000000000000-222-20000    orthogonal lifted from D4
ρ182-222-2-22-200000000000022-2-20000    orthogonal lifted from D4
ρ192-22-22-2-2200000000-2-20000002200    orthogonal lifted from D4
ρ20222-2-22-2-20000000000-2-200000022    orthogonal lifted from D4
ρ212-2-2-2-222200000000002-2000000-22    orthogonal lifted from D4
ρ2222-2-22-22-200000000-220000002-200    orthogonal lifted from D4
ρ232-222-2-22-2000000000000-2-2220000    orthogonal lifted from D4
ρ2422-22-2-2-220000000000002-2-220000    orthogonal lifted from D4
ρ2522-2-22-22-2000000002-2000000-2200    orthogonal lifted from D4
ρ26222-2-22-2-2000000000022000000-2-2    orthogonal lifted from D4
ρ272-22-22-2-220000000022000000-2-200    orthogonal lifted from D4
ρ282-2-2-2-22220000000000-220000002-2    orthogonal lifted from D4

Smallest permutation representation of C2×C41D4
On 32 points
Generators in S32
(1 12)(2 9)(3 10)(4 11)(5 24)(6 21)(7 22)(8 23)(13 18)(14 19)(15 20)(16 17)(25 32)(26 29)(27 30)(28 31)
(1 2 3 4)(5 6 7 8)(9 10 11 12)(13 14 15 16)(17 18 19 20)(21 22 23 24)(25 26 27 28)(29 30 31 32)
(1 18 31 6)(2 19 32 7)(3 20 29 8)(4 17 30 5)(9 14 25 22)(10 15 26 23)(11 16 27 24)(12 13 28 21)
(1 26)(2 25)(3 28)(4 27)(5 24)(6 23)(7 22)(8 21)(9 32)(10 31)(11 30)(12 29)(13 20)(14 19)(15 18)(16 17)

G:=sub<Sym(32)| (1,12)(2,9)(3,10)(4,11)(5,24)(6,21)(7,22)(8,23)(13,18)(14,19)(15,20)(16,17)(25,32)(26,29)(27,30)(28,31), (1,2,3,4)(5,6,7,8)(9,10,11,12)(13,14,15,16)(17,18,19,20)(21,22,23,24)(25,26,27,28)(29,30,31,32), (1,18,31,6)(2,19,32,7)(3,20,29,8)(4,17,30,5)(9,14,25,22)(10,15,26,23)(11,16,27,24)(12,13,28,21), (1,26)(2,25)(3,28)(4,27)(5,24)(6,23)(7,22)(8,21)(9,32)(10,31)(11,30)(12,29)(13,20)(14,19)(15,18)(16,17)>;

G:=Group( (1,12)(2,9)(3,10)(4,11)(5,24)(6,21)(7,22)(8,23)(13,18)(14,19)(15,20)(16,17)(25,32)(26,29)(27,30)(28,31), (1,2,3,4)(5,6,7,8)(9,10,11,12)(13,14,15,16)(17,18,19,20)(21,22,23,24)(25,26,27,28)(29,30,31,32), (1,18,31,6)(2,19,32,7)(3,20,29,8)(4,17,30,5)(9,14,25,22)(10,15,26,23)(11,16,27,24)(12,13,28,21), (1,26)(2,25)(3,28)(4,27)(5,24)(6,23)(7,22)(8,21)(9,32)(10,31)(11,30)(12,29)(13,20)(14,19)(15,18)(16,17) );

G=PermutationGroup([(1,12),(2,9),(3,10),(4,11),(5,24),(6,21),(7,22),(8,23),(13,18),(14,19),(15,20),(16,17),(25,32),(26,29),(27,30),(28,31)], [(1,2,3,4),(5,6,7,8),(9,10,11,12),(13,14,15,16),(17,18,19,20),(21,22,23,24),(25,26,27,28),(29,30,31,32)], [(1,18,31,6),(2,19,32,7),(3,20,29,8),(4,17,30,5),(9,14,25,22),(10,15,26,23),(11,16,27,24),(12,13,28,21)], [(1,26),(2,25),(3,28),(4,27),(5,24),(6,23),(7,22),(8,21),(9,32),(10,31),(11,30),(12,29),(13,20),(14,19),(15,18),(16,17)])

C2×C41D4 is a maximal subgroup of
C42.413D4  C42.82D4  C24.24D4  C42.432D4  C42.112D4  M4(2)⋊12D4  C42.118D4  C429D4  (C2×C4)⋊2D8  (C2×C8)⋊20D4  C4⋊C47D4  C4214D4  C24.219C23  C23.262C24  C23.328C24  C24.263C23  C23.333C24  C23.345C24  C4220D4  C4221D4  C42.171D4  C4227D4  C42.194D4  C4231D4  C23.569C24  C23.573C24  C24.411C23  C4233D4  C4247D4  C4312C2  C4315C2  C42.444D4  C42.240D4  M4(2)⋊7D4  C42.263D4  C42.266D4  C42.275D4  C2×D42  C22.87C25  C22.97C25  C22.132C25
C2×C41D4 is a maximal quotient of
C4216D4  C23.333C24  C23.401C24  C23.439C24  C4219D4  C4220D4  C42.167D4  C23.556C24  C4231D4  C42.196D4  C4247D4  C4312C2  C4315C2  C4219Q8  C42.360D4  C42.247D4  M4(2)⋊7D4  M4(2)⋊8D4  M4(2)⋊9D4  M4(2)⋊10D4  M4(2)⋊11D4  M4(2).20D4

Matrix representation of C2×C41D4 in GL5(ℤ)

-10000
01000
00100
000-10
0000-1
,
10000
0-1200
0-1100
00001
000-10
,
-10000
01000
00100
00001
000-10
,
10000
0-1000
0-1100
000-10
00001

G:=sub<GL(5,Integers())| [-1,0,0,0,0,0,1,0,0,0,0,0,1,0,0,0,0,0,-1,0,0,0,0,0,-1],[1,0,0,0,0,0,-1,-1,0,0,0,2,1,0,0,0,0,0,0,-1,0,0,0,1,0],[-1,0,0,0,0,0,1,0,0,0,0,0,1,0,0,0,0,0,0,-1,0,0,0,1,0],[1,0,0,0,0,0,-1,-1,0,0,0,0,1,0,0,0,0,0,-1,0,0,0,0,0,1] >;

C2×C41D4 in GAP, Magma, Sage, TeX

C_2\times C_4\rtimes_1D_4
% in TeX

G:=Group("C2xC4:1D4");
// GroupNames label

G:=SmallGroup(64,211);
// by ID

G=gap.SmallGroup(64,211);
# by ID

G:=PCGroup([6,-2,2,2,2,-2,2,217,103,650,158]);
// Polycyclic

G:=Group<a,b,c,d|a^2=b^4=c^4=d^2=1,a*b=b*a,a*c=c*a,a*d=d*a,b*c=c*b,d*b*d=b^-1,d*c*d=c^-1>;
// generators/relations

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Character table of C2×C41D4 in TeX

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